Abstract:
This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.
Keywords:
vortices on a torus, vortex lattices, point vortices, nonintegrability, chaos, invariant manifold, Poincaré map, topological analysis, numerical analysis, accuracy of calculations, reduction, reduced system.
The work of A. A. Kilin was supported by RFBR under the scientific project No. 17-01-00846-a. The work of E. M. Artemova was supported by RFBR under the scientific project No. 18-29-10050-mk.
\Bibitem{KilArt19}
\by Alexander A. Kilin, Lizaveta M. Artemova
\paper Integrability and Chaos in Vortex Lattice Dynamics
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 1
\pages 101--113
\mathnet{http://mi.mathnet.ru/rcd392}
\crossref{https://doi.org/10.1134/S1560354719010064}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061080675}
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This publication is cited in the following 7 articles:
Clodoaldo Grotta-Ragazzo, Björn Gustafsson, Jair Koiller, “On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d”, Regul. Chaotic Dyn., 29:2 (2024), 241–303
A. A. Kilin, E. M. Artemova, “Bifurcation Analysis of the Problem of Two Vortices on a Finite Flat Cylinder”, Rus. J. Nonlin. Dyn., 20:1 (2024), 95–111
E. M. Artemova, “Dinamika dvukh vikhrei na konechnom ploskom tsilindre”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:4 (2023), 642–658
Vikas S. Krishnamurthy, Takashi Sakajo, “The N-vortex problem in a doubly periodic rectangular domain with constant background vorticity”, Physica D: Nonlinear Phenomena, 448 (2023), 133728
Klas Modin, Milo Viviani, “Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey”, Arnold Math J., 7:3 (2021), 357
F. Grotto, “Essential self-adjointness of Liouville operator for 2D Euler point vortices”, J. Funct. Anal., 279:6 (2020), 108635
I. A. Bizyaev, I. S. Mamaev, “Dynamics of a pair of point vortices and a foil with parametric excitation in an ideal fluid”, Vestn. Udmurt. Univ.-Mat. Mekh. Kompyuternye Nauk., 30:4 (2020), 618–627
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